Mmmmm, electrical faults are are always a problem to source.
Things to find out.
When running take a volt reading across the battery, if it is greater than 12.5 volts, chances are the it is charging and the battery is knackered. an old battery may charge fully on a mains charger but the bike can not put enough current in to re-charge.
The Haynes manual chapter 6 has a good(ish) guide on the electrical system.
If the bike starts with a flat battery on the kickstart with the lights off, the alternator is most probalbly fine.
Have fun and let us know what the problem was when you sort it.